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Poisson limits for a clustering model of strauss

Published online by Cambridge University Press:  14 July 2016

Roy Saunders  and
Gerald M. Funk
Roy Saunders
Affiliation:
Northern Illinois University
Gerald M. Funk
Affiliation:
Northern Illinois University
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Abstract

In this article we present a limiting result for the random variable Yn(r) which arises in a clustering model of Strauss (1975). The result is that under some sparseness-of-points conditions the process {Yn(r): 0 ≦ r ≦ r∞} converges weakly to a non-homogeneous Poisson process {Y(r): 0 ≦ r ≦ r∞} when n → ∞. Simulation results are given to indicate the accuracy of the approximation when n is moderate and applications of the limiting result to tests for clustering are discussed.

Keywords

CLUSTERING MODELPOISSON PROCESSUNIFORM DISTRIBUTIONWEAK CONVERGENCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 4 , December 1977 , pp. 776 - 784
Copyright
Copyright © Applied Probability Trust 1977 

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Footnotes

This work has been partially supported by National Science Foundation Grant MCS 77–03582.

References

Billingsley, P. (1968) Weak Convergence of Probability Measures. Wiley, New York.Google Scholar
Kelly, F. P. and Ripley, B. D. (1976) A note on Strauss's model for clustering. Biometrika 63, 354–360.Google Scholar
Pyke, R. (1959) The supremum and infimum of the Poisson process. Ann. Math. Statist. 30, 568–576.Google Scholar
Saunders, R. and Funk, G. M. (1976) Convergence of a process arising in a clustering model. Unpublished technical report.Google Scholar
Strauss, D. J. (1975) A model for clustering. Biometrika 62, 467–475.Google Scholar